Structured distance to normality of tridiagonal matrices

In this article we study the distance d, measured in the Frobenius norm, of a tridiagonal matrix T to the set IT of similarly structured irreducible normal matrices. The matrices in the closure of IT whose distance to T is d are characterized. Known results in the literature for the cases in which T is real or a Toeplitz matrix are recovered. In addition, the special case in which T is a 2-Toeplitz matrix is considered.